On Certain Threshold Copulas Estimators
Evgueni Haroutunian
Irina Safaryan
Institute for Informatics and Automation Problems
of NAS of RA
Institute for Informatics and Automation Problems
of NAS of RA
e-mail: evhar@ipia.sci.am
e-mail: safari@ipia.sci.am
ABSTRACT
A method of construction of empirical associate measures for
components of two-dimensional random vector in threshold
copula models is proposed. It is shown that for such models
Spearman’s rank correlation coefficient and Kendall’s concordance coefficient are the linear functions of two-sample
Wilcoxon statistics.
two-dimensional DF with marginal DF’s by relationship
F (x, y) = C(FX (x), FY (y)).
The concordance function K(C1 , C2 ), defined by Nelsen in
[3], denotes the difference between the probabilities of concordance and disconcordance of random vectors (X1 , Y1 ) and
(X2 , Y2 ) with copulas C1 (u, v), C2 (u, v), i.e.
K(C1 , C2 ) = Pr((X1 − X2 )(Y1 − Y2 ) > 0)−
Keywords
Copula, empirical associate measures, threshold dependence
model, Kendall’s concordance coefficient, Spearman’s rank
correlation
−Pr((X1 − X2 )(Y1 − Y2 ) < 0).
It can be presented in the form
Z Z
K(C1 , C2 ) = 4
C2 (u, v)dC1 (u, v) − 1,
(1)
D
1. INTRODUCTION
In two-dimensional data analysis situations arise when rather
high values of certain empirical measures of association such
as Spearman’s rank correlation and Kendall’s concordance
cannot be interpreted as monotone dependence between components of two-dimensional random vector. In particular,
this refers to threshold dependence structures in which one
of the components serves as categorizing variable for the
other. In case under consideration the observed sequence
can be divided on two or more groups concentrating along
some line on plane, while correlation between components
inside each group is missed.
or in an equivalent form as
Z Z
∂C2 (u, v) ∂C1 (u, v)
K(C1 , C2 ) = 1 − 4
dudv,
∂u
∂u
D
(2)
where D = [0, 1] × [0, 1] is the unique square.
Then, as it is shown in [3] many measures of association
between random variables (RV’s) X and Y whose copula is
C can be defined in terms of concordance function. For
instance, Spearmans correlation coefficient ρC is proportional to concordance function with arguments C1 = C and
C2 = Π = uv, that is
ρC = 3K(C, Π).
The separation of two, or many groups is based on quantiles of the categorizing variable, called thresholds which are
not known in general. To obtain consistent estimates of unknown threshold in one-threshold model Safaryan, Haroutunian and Manasyan in [1] applied the change-point detection
technique. The representation of dependence of two dimensional random vector components by copulas was derived
in [2]. We propose a method of construction of empirical
measures for two-dimensional dependence structures with
applying the results obtained in [2] as well as the function
of concordance between two copulas defined by Nelsen in
[3] and empirical copula notion introduced by Fermanian,
Radulovic and Wegkamp in [4].
and Kendall’s concordance coefficient τC is equal to K(C, C)
Thus taking in account (1) well-known representation for ρC
and τC can be obtained, namely,
2. ASSOCIATION MEASURES FOR ONETHRESHOLD COPULA MODELS
Definition 1: A copula Cp (u, v) depending on scalar parameter p ∈ (0, 1), is called one–threshold if u = p is a single
point for which the following relations hold
Let (X, Y ) be a random vector with two-dimensional distribution function (DF) F (x, y), continuous marginal DF’s
FX (x) and FY (y) and corresponding copula C(u, v). We
remind that copula C(u, v) is a function, which connects
Z1 Z1
ρC = 12
0
(C(u, v) − uv)dudv
(3)
C(u, v)dC(u, v) − 1.
(4)
0
and
Z1 Z1
τC = 12
0
0
We introduce notion of threshold copulas and derive corresponding expressions for ρC and τC .
Cp (u, v)
Cp (p, v)
=
,
v
p
u ≤ p,
v − Cp (u, v)
v − Cp (p, v)
=
,
1−u
1−p
and
Cp (u, v) 6= pv.
u > p,
Theorem 1: Spearman’s correlation coefficient ρCp for onethreshold copula Cp is the following:
Z1
ρCp = 6
Cp (p, v)dv − 3p.
(5)
C(u, v) in (3) with ĈN (u, v) given in (8) we obtain Spearman’s rank correlation coefficient between RV’s X and Y ,
in the form
N
X
12
N +1
N +1
(RXn −
)(RYn −
).
2
N (N + 1) n=1
2
2
ρ̂C =
0
For Kendall’s concordance coefficient the following relation
is true
2
τCp = ρCp .
3
Proof: We note that definition of one-threshold copula completely coincide with definitions of the threshold dependence
between RV’s X and Y expressed in terms of conditional distributions of RV Y under conditions {X ≤ µ}, {X > µ},
brought in [3] if p = FX (µ). Consequently, Cp can be represented as follows
1
Cp (u, v) = uv +
(C(p, v) − pv)(min(u, p) − pu). (6)
p(1 − p)
The further proof immediately follows by substitution of (6)
in (3) and (4) and integration of the derived quintile.
The obtained expressions for Spearmann’s and Kendall’s coefficients allow to construct estimators for τC and ρC which
are based only on ranks of Y and propose a new algorithm
for the estimation of parameter p .
3. ESTIMATORS FOR ONE-THRESHOLD
DEPENDENCE CASE
The sample versions of measures of association can be described by empirical copula function ĈN (u, v), which has
been established by Fermanian, Radulovic and Wegkamp in
[4].
Let {(Xn , Yn )N
n=1 } be a random sample from RV (X, Y ),
with DF F (x, y), marginal DF’s FX (x) and FY (y) and copula C(u, v). Consider the following empirical functions:
F̂N,X (x) = #{Xn : Xn ≤ x} =
N
1 X
1{Xn ≤ x},
N n=1
N
1 X
1{Yn ≤ y},
N n=1
N
1 X
1{Xn ≤ x} × 1{Yn ≤ y}.
N n=1
N
1 X
1{F̂N,X (Xn ) ≤ u, F̂N,Y (Yn ) ≤ v}. (7)
N n=1
N
1 X
RYn
R Xn
≤ u} × 1{
≤ v},
1{
N n=1 N + 1
N +1
(9)
0
Where {Yn , }N
n=1 is the of induced order statistics sequence,
0
(i.e. for X(1) < X(2) < < X(N ) induced statistic Yn is
0
defined as Yn = Yi , if X(n) = Xi .
Proof:
Z1
ρ̂cp = 6
ĈN (p, v)dv − 3p =
0
N X
N
X
RY j
6
1{RXn ≤ p} × 1{RYn ≤
} − 3p =
N (N + 1) n=1 j=1
N +1
=
[N p] N
XX
6
1{RY 0 ≤ RYj } − 3p.
n
N (N + 1) n=1 j=1
The last expression proves (9).
Let
WN (
n
n
N
1 X RYi0
1
)=
(
− ), n = 1, N − 1, (10)
N
N − n n i=1 N + 1
2
6(N − n(p))n(p)
n(p)
n(p) − N p
WN (
) + 3(
). (11)
N2
N
N
We use (11) to estimate ρcp if n(p) is known, otherwise a
change point technique proposed in [1] can be applied
A representation of empirical copula suggested in [5] is the
following
ĈN (u, v) =
[N p]
X
6
(N + 1 − RY 0 ) − 3p,
n
(N + 1)N n=1
ρ̂cp = −
The cadlag version of empirical copula, according to [4] is
defined as follows
ĈN (u, v) =
ρ̂Cp =
Corollary 1: Rank correlation coefficient of Spearman is
the linear function from Wilcoxon statistic WN ( n(p)
), n(p) =
N
[pN ] defined by relation
F̂N (x, y) = #{(Xn , Yn ) : Xn ≤ x, Yn ≤ y} =
=
Theorem 2: Expression of Spearman’s rank coefficients for
one-threshold model are the following:
is the sequence of Wilcoxon statistic, which tests homogeinety
0
0
N
of two samples, {Yi }n
i=1 and {Yi }i=n+1 .
Then from Theorem 2 can be deduced the following consequences:
where 1 {A} is the indicator of event A. Let
F̂N,Y (y) =
For the threshold models we obtain sampling estimates of
ρCp by substituting (8) in (5). Then the following theorem
holds.
(8)
where RXn and RYn are ranks of RV’s Xn and Yn in seN
quences {Xn }N
n=1 and {Yn }n=1 respectively. After replacing
Let
n̂ = arg max |WN (
0<n<N
n
)|
N
and
r
n̂
n̂
= WN ( ) 12(1 − )n̂.
N
N
Then, a consisent estimate of ρcp is defined in the following
∗
WN
∗
∗
Corollary 2: If WN
> zα , or WN
< 1 − zα , where zα is
quantile of level α of RVZ distributed as N (0, 1), then the
consistent estimate of p is defined by
p̂ =
n̂
,
N
and consistent estimator of ρCp is the following
ρ̂Cp = −6(1 − p̂)p̂WN (p̂).
Theorem 4. Expression of Spearman’s rank coefficient for
two-threshold model is the following:
Proof: The induced order statistic sequence {Yn,X }N
n=1
can be viewed as a random sample from conditional DF
F (y|x) = P r{Y ≤ y|X = x}. As it follows from Definition
1 that probabilities
0
P r{Yn ≤ y|X = F −1 (p)},
The proof is similar of the proof of Theorem 1.
n = 1, N ,
are the same if n ≤ [N p] and the other if n > [N p] then the
number n(p) = [N p] can be called the shangepoint for the
0
sequence {Yn }N
n=1 .
ρ̂cp =
[N p1 ]
X
6
(
p2 (N + 1 − RY 0 )+
n
(N + 1)N n=1
[N p2 ]
+
X
(1 − p1 )(N + 1 − RY 0 − 3p2 )
n=1
n
4. ESTIMATORS FOR TWO-THRESHOLD
DEPENDENCE CASE
The further extension of the noted results are connected
with empirical threshold copulas of the K-dimensional random vector X (1) , ..., X (K) in the case, when RV X (1) serves
categorizing variable for X (2) , ..., X (K) .
Similar results can be obtained for cases of two and more
thresholds.
REFERENCES
Definition 2: A copula Cp depending on vector parameter
p = (p1 , p2 ), 0 < p1 < p2 1 < 1 is called two-threshold is
there exist exactly two values u = p1 and u = u2 such that
relations holds
Cp (p1 , v)
Cp (u, v)
=
,
u
p1
f or
u ≤ p1 ,
Cp (u, v) − Cp (p1 , v)
Cp (p2 , v) − Cp (p1 , v)
=
,
u − p1
p2 − p1
f or
p1 < u ≤ p2 ,
v − Cp (u, v)
v − Cp (p2 , v)
=
, f or u > p2 ,
1−u
1 − p2
and
p2 Cp (p1 , v) 6= p1 Cp (p2 , v),
(p2 − p1 ) 6= p1 Cp (p2 , v).
Theorem 3: Spearman’s correlation coefficient ρcp for twothreshold copula Cp is equal to
Z1
ρcp = 6
p2 Cp (p1 , v) + (1 − p1 )Cp (p2 , v) − 3p2 .
0
For Kendall’s concordance coefficients the following relation
is true
2
τcp = ρcp .
3
Proof: We note that dependence between RV’s X and Y expressed in terms of conditional distributions of RV Y under
conditions {X ≤ µ1 }, {X > µ2 }, {µ1 < X ≤ µ2 } brougth
in [3], if p1 = FX (µ1 ), p2 = FX (µ2 ). can be represented by
Cp as follows
Cp (u, v) = uv + ∆1 (v, p)(min(u, p1 ) − p1 u)+
∆1 (v, p)(min(u, p2 ) − p2 u),
where
∆1 =
1
(p2 (Cp (p1 , v) − p1 v) − p1 (Cp (p2 , v) − p2 v)),
p1 (p2 − p1 )
and
∆2 =
(1 − p1 )(Cp (p1 , −p2 v) − (1 − p2 )(Cp (p1 , v) − p1 v))
(1 − p2 )(p2 − p1 )
[1] E. Haroutunian, I. Safaryan and A. Manasyan,
“Two-dimentsional sequence homogeneity testing
against mixture alternative”, Mathematical Problems
of Computer Science, vol. 23, pp. 67–79, 2004.
[2] E. Haroutunian and I. Safaryan,“Copulas of
two-dimensional threshold models”, Mathematical
Problems of Computer Science, vol. 31, pp. 40–48,
2008.
[3] B. R. Nelsen,“Concordance and copulas”, The Annals
of Statistics, vol. 9, no. 6, pp. 879–886, 1981.
[4] J. D. Fermanian, D. Radulovic and M. Wegkamp,
“Weak convergence of empirical copula processes”,
Bernoulli, vol. 10, no. 5, pp. 847–860, 2004.
[5] B. Remillard and O. Scaillet, “Testing for equality
between two copulas”, Journal of Multivariate
Analysis, vol. 100, no. 3, pp. 377– 386, 2008.